A165570 Successively better golden semiprimes.

6, 15, 77, 589, 851, 1363, 15229, 201563, 512893, 644251, 1366553, 3416003, 7881197, 377331139, 400711231, 2963563859, 4035221017, 28862500577, 52027213697, 133793658289, 418298061641, 1363588753103, 1970239102459, 6355462656397, 136388198153719, 465337655023099

Offset

1,1

Comments

This is lexicographically earliest sequence of such semiprimes p*q, starting from 6=2*3, that for each successive term p*q, q/p is a better approximant of Golden ratio (1+sqrt(5))/2 than the previous term. See A165569 for the exact procedure.

Can it be proved that this a subset of A108540?

The ratio A165572(n)/A165571(n) converges towards golden ratio = (1+sqrt(5))/2 = 1.618033988749895... as: 1.5, 1.6666666666666667, 1.5714285714285714, 1.631578947368421, 1.608695652173913, 1.6206896551724137, 1.6185567010309279, 1.6175637393767706, 1.6181172291296626, 1.618066561014263, 1.618063112078346, 1.618031658637302, 1.6180335296782964, 1.6180341824372995, 1.6180339327699054, ...

Links

Amiram Eldar, Table of n, a(n) for n = 1..48

Formula

a(n) = A165571(n)*A165572(n) = A000040(A165569(n))*A108539(A165569(n)).

Mathematica

f[p_] := Module[{x = GoldenRatio * p, p1, p2}, p1 = NextPrime[x, -1]; p2 = NextPrime[p1]; If[p2 - x > x - p1, p1, p2]]; seq={}; dm = 1; p1 = 1; Do[p1 = NextPrime[p1]; k++; p2 = f[p1]; d = Abs[p2/p1 - GoldenRatio]; If[d < dm, dm = d; AppendTo[seq, p1*p2]], {10^4}]; seq (* Amiram Eldar, Nov 28 2019 *)

Crossrefs

Cf. A108539, A165569, A165571, A165572.

Sequence in context: A177122 A108540 A232170 * A260117 A220360 A096565

Adjacent sequences: A165567 A165568 A165569 * A165571 A165572 A165573

Keyword

nonn

Author

Antti Karttunen, Sep 22 2009

Extensions

a(16)-a(23) from Donovan Johnson, May 13 2010

a(24)-a(26) from Amiram Eldar, Nov 28 2019

Status

approved